We call the geometric-harmonic mean of numbers \(a\) and \(b\) the general limit of the sequences \(\{a_n\}\) and \(\{b_n\}\) constructed according to the rule \(a_0 = a\), \(b_0 = b\), \(a_{n + 1} = \frac{2a_nb_n}{a_n + b_n}\), \(b_{n + 1} = \sqrt{a_nb_n}\) (\(n \geq 0\)).
We denote it by \(\nu (a, b)\). Prove that \(\nu (a, b)\) is related to \(\mu (a, b)\) (see problem number 61322) by \(\nu (a, b) \times \mu (1/a, 1/b) = 1\).
Problem number 61322 says that both of these sequences have the same limit.
This limit is called the arithmetic-geometric mean of the numbers \(a, b\) and is denoted by \(\mu (a, b)\).
A class contains 33 pupils, who have a combined age of 430 years. Prove that if we picked the 20 oldest pupils they would have a combined age of no less than 260 years. The age of any given pupil is a whole number.
In the first term of the year Daniel received five grades in mathematics with each of them being on a scale of 1 to 5, and the most common grade among them was a 5 . In this case it turned out that the median of all his grades was 4, and the arithmetic mean was 3.8. What grades could Daniel have?
Gabby and Joe cut rectangles out of checkered paper. Joe is lazy; He throws a die once and cuts out a square whose side is equal to the number of points that are on the upwards facing side of the die. Gabby throws the die twice and cuts out a rectangle with the length and width equal to the numbers which come out from the die. Who has the mathematical expectation of the rectangle of a greater area?
According to the rules of a chess match, the winner is declared to be the one who has beaten their opponent by two defeats. Draws do not count. The probability of winning for both rivals is the same. The number of successful games played in such a match is random. Find its mathematical expectation.
A die is thrown six times. Find the mathematical expectation of the number of different faces the die lands on.
A sailor can only serve on a submarine if their height does not exceed 168 cm. There are four teams \(A\), \(B\), \(C\) and \(D\). All sailors in these teams want to serve on a submarine and have been rigorously selected. There remains the last selection round – for height.
In team \(A\), the average height of sailors is 166 cm.
In team \(B\), the median height of the sailors is 167 cm.
In team \(C\), the tallest sailor has a height of 169 cm.
In team \(D\), the mode of the height of the sailors is 167 cm.
In which team, can at least half of the sailors definitely serve on the submarine?
The length of the hypotenuse of a right-angled triangle is 3.
a) The Scattered Scientist calculated the dispersion of the lengths of the sides of this triangle and found that it equals 2. Was he wrong in the calculations?
b) What is the smallest standard deviation of the sides that a rectangular triangle can have? What are the lengths of its sides, adjacent to the right angle?
The upper side of a piece of square paper is white, and the lower one is red. In the square, a point F is randomly chosen. Then the square is bent so that one randomly selected vertex overlaps the point F. Find the mathematical expectation of the number of sides of the red polygon that appears.
The Scattered Scientist constructed a device consisting of a sensor and a transmitter. The average life expectancy of the sensor part is 3 years, the average lifetime of the transmitter is 5 years. Knowing the distribution of the lifetime of the sensor and the transmitter, the Scattered Scientist calculated that the average lifetime of the entire device is 3 years 8 months. Was the Scattered Scientist wrong in his calculations?